Hypergraphs [1]}, [2]}, [3]} naturally encode higher-order interactions in terms of fundamental units, their hyperedges. However, in many cases the data only reveals pairwise interactions making the simple graph the appropriate representation and the higher order interactions have to be inferred from the pairwise interactions recorded in a network. For instance, data on phone calls does not reveal other higher-order connections, e.g. through face-to-face meetings, but the pattern of calls can reveal the existence of such connections.
Network Science offers many approaches to discovering these local group interactions from a network.
It is natural to work with small sub-graphs, motifs [4]}, [5]}, graphlets [6]}, [7]}. Higher order analysis in terms of paths is important in several contexts[8]}, [9]} but more relevant here is the use of cliques, subgraphs which are complete graphs. Cliques have long played an important role in social science, for instance see [10]}, and can be used in many contexts such as community detection [11]}, [12]}. Cliques in networks are the basis for analysis in terms of simplicial complexes as used in algebraic topology. In network analysis, simplicial complexes [13]}, [14]}, [15]} have been used to analyse network geometry [16]}, to model structure in temporal networks [17]}, investigate synchronization phenomenon [18]}, [19]}, [20]}, [21]}, social contagion [22]}, epidemic spreading [23]}, and neuroscience [24]}, [25]}.
